Periodic and quasi-periodic behavior in resource-dependent age structured population models
- 75 Downloads
To describe the dynamics of a resource-dependent age structured population, a general non-linear Leslie type model is derived. The dependence on the resources is introduced through the death rates of the reproductive age classes. The conditions assumed in the derivation of the model are regularity and plausible limiting behaviors of the functions in the model. It is shown that the model dynamics restricted to its ω-limit sets is a diffeomorphism of a compact set, and the period-1 fixed points of the model are structurally stable. The loss of stability of the non-zero steady state occurs by a discrete Hopf bifurcation. Under general conditions, and after the loss of stability of the structurally stable steady states, the time evolution of population numbers is periodic or quasi-periodic. Numerical analysis with prototype functions has been performed, and the conditions leading to chaotic behavior in time are discussed.
Unable to display preview. Download preview PDF.
- Arnold, V. I., V. S. Afrajmovich, Yu. S. Ilyashenko and L. P. Shilnikov (1991). I. Bifurcation theory, in Encyclopaedia of Mathematical Sciences, Dynamical Systems, Vol. 5, Berlin: Springer-Verlag.Google Scholar
- Arrowsmith, D. K. and C. M. Place (1991). An Introduction to Dynamical Systems, Cambridge: Cambridge University Press.Google Scholar
- Bailey, N. T. J. (1964). The Elements of Stochastic Processes, New York: Wiley Interscience Publications.Google Scholar
- Caswell, H. (1989). Matrix Population Models, Sunderland (MA): Inc. Publishers.Google Scholar
- Cushing, J. M. (1998). An Introduction to Structured Population Dynamics, Philadelphia: SIAM.Google Scholar
- Guckenheimer, J. and P. Holmes (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Berlin: Springer-Verlag.Google Scholar
- Kuznetsov, Y. A. (1995). Elements of Applied Bifurcation Theory, New York: Springer-Verlag.Google Scholar
- MacArthur, R. H. (1972). Geographical Ecology, Princeton: Princeton University Press.Google Scholar
- May, R. M. (1987). Chaos and the dynamics of biological populations, in Dynamical Chaos, M. V. Berry, I. C. Percival and N. O. Weiss (Eds), Princeton: Princeton University Press, pp. 27–44.Google Scholar
- Oster, G. and J. Guckenheimer (1976). Bifurcation phenomena in population models, in The Hopf Bifurcation and its Applications, J. E. Marsden and M. McCracken (Eds), New York: Springer-Verlag.Google Scholar
- Ruelle, D. (1989). Elements of Differentiable Dynamics and Bifurcation Theory, Boston: Academic Press.Google Scholar
- Wilbur, H. M. (1996). Multistage life cycles, in Population Dynamics in Space and Time, O. E. Rhodes Jr, R. K. Chesser and M. H. Smith (Eds), Chicago: University of Chicago Press.Google Scholar